Following the theoretical studies of J.B.~Robinson and H.W.~Kuhn in the late 1940s and the early 1950s, G.B.~Dantzig, R.~Fulkerson, and S.M.~Johnson demonstrated in 1954 that large instances of the TSP could be solved by linear programming. Their approach remains the only known tool for solving TSP instances with more than several hundred cities; over the years, it has evolved further through the work of M.~Gr\"otschel, S.~Hong, M.~J\"unger, P.~Miliotis, D.~Naddef, M.~Padberg, W.R.~Pulleyblank, G.~Reinelt, G.~Rinaldi, and others. We enumerate some of its refinements that led to the solution of a 13,509-city instance.